Problem: Furnace came on as usual in the morning but failed to shut off after the set temp on the thermostat was reached. The house temp (as measured on the thermostat) reached about 10 degrees higher than the set temp and hot air was still blowing out of the registers.

Thermostat: Inexpensive (Honeywell), programmable thermostat

Heating system: Hydronic Air Handler? In case I'm getting the name wrong, the basic setup is that the Hot Water heater (same as for hot tap water) sends hot water to the 'furnace' where a fan blows over a heat exchanger thus blowing hot air out through the ducts. (Sorry I don't have the brand and model/info of the furnace unit handy right now - it is about 17 years old.)

Trouble-shooting/Notes:
Switching the thermostat switch to System-Off (vs Heat or Cool) did not turn off the hot air coming out of the registers (i.e. 'burner' and blower still On).

Fan switch on Thermostat was correctly set on Auto. Switching fan from Auto to On, waiting, and then setting back to Auto DID turn the furnace and blower Off. Like it 'alerted' the furnace to the fact that the temp had been reached/exceeded and it could now turn off. (This is the weirdest part, I haven't seen this same symptom listed elsewhere online.)

I replaced the Thermostat with a similar one and the problem still recurred - after switching the system off during the thermostat replacement and it being off as expected until the 'wake-up' program time, the system failed to shut off (even when exceeding the set temperature by 10-degrees). The system did shut off with the same 'trick' of switching the fan from Auto to On, and back to Auto.

The posts I've found so far online mention Limit Switch (which seems to be relevant only if Cold air were coming out) and the Furnace Relay Switch. It's not clear to me if the problem IS the Furnace Relay Switch if fixing it is something I'd be able to handle as a novice DIYer.

I'd appreciate any feedback on what might be causing this problem, on further trouble-shooting or testing that I should do, and of course on any beginner-level user-serviceable types of solutions.

The furnace or a short in the wires to the thermostat I have seen the wires short from nails driven into them, mice chewing on them and people walking on the cable in the attic. Even with 3 causes I have seen it is probably in the furnace.
– Ed BealDec 27 '17 at 23:05

I would check the heat anticipator in the thermostat. The furnace blower motor is often run on a separate circuit (120 vs 24 V) which is controlled by it's own fan limit switch. This switch keeps the blower running after the call for heat has been satisfied in order to extract the heat from the furnace, which is why you will feel heated air from the registers.

It sounds like the furnace is operational, however you are having a problem with "overshoot". It may not necessarily be a faulty thermostat though, it may simply need adjusting. Google has plenty of information on how to do that as it's fairly simple.

This is only my opinion after a quick read-through, so take it for what it's worth.

According to a furnace repair person who came out today, the most likely culprits are failing pump relay or fan relay. (Though of course Murphy's Law applied and the problem did not manifest for the tech.)

Why is my hydronic furnace heating the house far past the thermostat setting? The heat exchanger is still being feed hot water, and the fan will never shut off until it gets cool enough, which it never will if it has a "failing pump relay". +1
– MazuraJan 24 '18 at 22:37